week 9*ExampleA device containing two key components fails when and only when both components fail. The lifetime, T1 and T2, of these components are independent with a common density function given by

The cost, X, of operating the device until failure is 2T1 + T2. Find the density function of X.

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week 9*ConvolutionSuppose X, Y jointly distributed random variables. We want to find the probability / density function of Z=X+Y.

Discrete case

X, Y have joint probability function pX,Y(x,y). Z = z whenever X = x and Y = z x. So the probability that Z = z is the sum over all x of these jointprobabilities. That is

If X, Y independent then

This is known as the convolution of pX(x) and pY(y).

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week 9*ExampleSuppose X~ Poisson(1) independent of Y~ Poisson(2). Find the distribution of X+Y.

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*Convolution - Continuous caseSuppose X, Y random variables with joint density function fX,Y(x,y). We want to find the density function of Z=X+Y.Can find distribution function of Z and differentiate. How?The Cdf of Z can be found as follows:

If is continuous at z then the density function of Z is given by

If X, Y independent then

This is known as the convolution of fX(x) and fY(y).

week 9*ExampleX, Y independent each having Exponential distribution with mean 1/. Find the density for W=X+Y.

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week 9*Some Recalls on Normal DistributionIf Z ~ N(0,1) the density of Z is

If X = Z + then X ~ N(, 2) and the density of X is

If X ~ N(, 2) then

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week 9*More on Normal DistributionIf X, Y independent standard normal random variables, find the density of W=X+Y.

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week 9*In general,

If X1, X2,, Xn i.i.d N(0,1) then X1+ X2++ Xn ~ N(0,n).

If , ,, then

If X1, X2,, Xn i.i.d N(, 2) then Sn = X1+ X2++ Xn ~ N(n, n2) and

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*Sum of Independent 2(1) random variablesRecall: The Chi-Square density with 1 degree of freedom is the Gamma( , ) density. If X1, X2 i.i.d with distribution 2(1). Find the density of Y = X1+ X2.

In general, if X1, X2,, Xn ~ 2(1) independent then X1+ X2++ Xn ~ 2(n) = Gamma(n/2, ).

Recall: The Chi-Square density with parameter n is

week 9*Cauchy DistributionThe standard Cauchy distribution can be expressed as the ration of two Standard Normal random variables.Suppose X, Y are independent Standard Normal random variables. Let . Want to find the density of Z.

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week 9*Change-of-Variables for Double IntegralsConsider the transformation , u = f(x,y), v = g(x,y) and suppose we are interested in evaluating .

Why change variables?In calculus: - to simplify the integrand. - to simplify the region of integration.In probability, want the density of a new random variable which is a function of other random variables.

Example: Suppose we are interested in finding . Further, suppose T is a transformation with T(x,y) = (f(x,y),g(x,y)) = (u,v). Then,

Question: how to get fU,V(u,v) from fX,Y(x,y) ?

In order to derive the change-of-variable formula for double integral, we need the formula which describe how areas are related under the transformation T: R2 R2 defined by u = f(x,y), v = g(x,y).

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week 9*JacobianDefinition: The Jacobian Matrix of the transformation T is given by

The Jacobian of a transformation T is the determinant of the Jacobian matrix.

In words: the Jacobian of a transformation T describes the extent to which T increases or decreases area.

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week 9*Change-of-Variable Theorem in 2-dimentionsLet x = f(u,v) and y = g(u,v) be a 1-1 mapping of the region Auv onto Axy with f, g having continuous partials derivatives and det(J(u,v)) 0 on Auv. If F(x,y) is continuous on Axy then

where

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week 9*Example Evaluate where Axy is bounded by y = x, y = ex, xy = 2 and xy = 3.

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week 9*Change-of-Variable for Joint DistributionsTheoremLet X and Y be jointly continuous random variables with joint density function fX,Y(x,y) and let DXY = {(x,y): fX,Y(x,y) >0}. If the mapping T givenby T(x,y) = (u(x,y),v(x,y)) maps DXY onto DUV. Then U, V are jointlycontinuous random variable with joint density function given by

where J(u,v) is the Jacobian of T-1 given by

assuming derivatives exists and are continuous at all points in DUV .

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week 9*ExampleLet X, Y have joint density function given by

Find the density function of

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week 9*ExampleShow that the integral over the Standard Normal distribution is 1.

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week 9*Density of QuotientSuppose X, Y are independent continuous random variables and we are interested in the density of

Can define the following transformation .

The inverse transformation is x = w, y = wz. The Jacobian of the inverse transformation is given by

Apply 2-D change-of-variable theorem for densities to get

The density for Z is then given by

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week 9*ExampleSuppose X, Y are independent N(0,1). The density of is

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week 9*Example F distributionSuppose X ~ 2(n) independent of Y ~ 2(m). Find the density of

This is the Density for a random variable with an F-distribution with parameters n and m (often called degrees of freedom). Z ~ F(n,m).

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week 9*Example t distributionSuppose Z ~ N(0,1) independent of X ~ 2(n). Find the density of

This is the Density for a random variable with a t-distribution with parameter n (often called degrees of freedom). T ~ t(n)

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week 9*Some Recalls on Beta DistributionIf X has Beta(,) distribution where > 0 and > 0 are positive parameters the density function of X is

If = = 1, then X ~ Uniform(0,1).If = = , then the density of X is

Depending on the values of and , density can look like:

If X ~ Beta(,) then and

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week 9*Derivation of Beta DistributionLet X1, X2 be independent 2(1) random variables. We want the density of

Can define the following transformation

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